G = C14.562+ 1+4 order 448 = 26·7
56th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.562+ 1+4, C4⋊C4⋊13D14, (C2×Q8)⋊7D14, C22⋊Q8⋊21D7, C28⋊7D4⋊46C2, C4⋊D28⋊27C2, (C2×D28)⋊8C22, C22⋊D28⋊17C2, D14⋊C4⋊33C22, (C2×C28).64C23, C4⋊Dic7⋊14C22, C22⋊C4.64D14, (Q8×C14)⋊10C22, Dic7⋊4D4⋊15C2, D14.5D4⋊22C2, C28.23D4⋊17C2, (C2×C14).188C24, Dic7⋊C4⋊20C22, C7⋊6(C22.32C24), (C4×Dic7)⋊30C22, (C22×C4).250D14, C2.38(D4⋊8D14), C2.58(D4⋊6D14), C22.4(Q8⋊2D7), (C22×D7).79C23, (C23×D7).55C22, C23.196(C22×D7), C22.209(C23×D7), (C22×C14).216C23, (C22×C28).316C22, (C2×Dic7).241C23, (C22×Dic7).124C22, (C2×C4×D7)⋊19C22, (C2×D14⋊C4)⋊27C2, C4⋊C4⋊D7⋊23C2, (C7×C4⋊C4)⋊22C22, (C7×C22⋊Q8)⋊24C2, C14.116(C2×C4○D4), C2.20(C2×Q8⋊2D7), (C2×C14).28(C4○D4), (C2×C4).185(C22×D7), (C2×C7⋊D4).40C22, (C7×C22⋊C4).43C22, SmallGroup(448,1097)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.562+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 1436 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23×D7, Dic7⋊4D4, C22⋊D28, D14.5D4, C4⋊D28, C4⋊C4⋊D7, C2×D14⋊C4, C28⋊7D4, C28.23D4, C7×C22⋊Q8, C14.562+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, Q8⋊2D7, C23×D7, D4⋊6D14, C2×Q8⋊2D7, D4⋊8D14, C14.562+ 1+4
►On 112 points
Generators in S112
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 61 26 84)(2 62 27 71)(3 63 28 72)(4 64 15 73)(5 65 16 74)(6 66 17 75)(7 67 18 76)(8 68 19 77)(9 69 20 78)(10 70 21 79)(11 57 22 80)(12 58 23 81)(13 59 24 82)(14 60 25 83)(29 95 55 102)(30 96 56 103)(31 97 43 104)(32 98 44 105)(33 85 45 106)(34 86 46 107)(35 87 47 108)(36 88 48 109)(37 89 49 110)(38 90 50 111)(39 91 51 112)(40 92 52 99)(41 93 53 100)(42 94 54 101)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 80)(58 81)(59 82)(60 83)(61 84)(62 71)(63 72)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 45 26 33)(2 44 27 32)(3 43 28 31)(4 56 15 30)(5 55 16 29)(6 54 17 42)(7 53 18 41)(8 52 19 40)(9 51 20 39)(10 50 21 38)(11 49 22 37)(12 48 23 36)(13 47 24 35)(14 46 25 34)(57 110 80 89)(58 109 81 88)(59 108 82 87)(60 107 83 86)(61 106 84 85)(62 105 71 98)(63 104 72 97)(64 103 73 96)(65 102 74 95)(66 101 75 94)(67 100 76 93)(68 99 77 92)(69 112 78 91)(70 111 79 90)
(1 40 8 33)(2 41 9 34)(3 42 10 35)(4 29 11 36)(5 30 12 37)(6 31 13 38)(7 32 14 39)(15 55 22 48)(16 56 23 49)(17 43 24 50)(18 44 25 51)(19 45 26 52)(20 46 27 53)(21 47 28 54)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 107 78 100)(72 108 79 101)(73 109 80 102)(74 110 81 103)(75 111 82 104)(76 112 83 105)(77 99 84 106)
G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,61,26,84)(2,62,27,71)(3,63,28,72)(4,64,15,73)(5,65,16,74)(6,66,17,75)(7,67,18,76)(8,68,19,77)(9,69,20,78)(10,70,21,79)(11,57,22,80)(12,58,23,81)(13,59,24,82)(14,60,25,83)(29,95,55,102)(30,96,56,103)(31,97,43,104)(32,98,44,105)(33,85,45,106)(34,86,46,107)(35,87,47,108)(36,88,48,109)(37,89,49,110)(38,90,50,111)(39,91,51,112)(40,92,52,99)(41,93,53,100)(42,94,54,101), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,45,26,33)(2,44,27,32)(3,43,28,31)(4,56,15,30)(5,55,16,29)(6,54,17,42)(7,53,18,41)(8,52,19,40)(9,51,20,39)(10,50,21,38)(11,49,22,37)(12,48,23,36)(13,47,24,35)(14,46,25,34)(57,110,80,89)(58,109,81,88)(59,108,82,87)(60,107,83,86)(61,106,84,85)(62,105,71,98)(63,104,72,97)(64,103,73,96)(65,102,74,95)(66,101,75,94)(67,100,76,93)(68,99,77,92)(69,112,78,91)(70,111,79,90), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,107,78,100)(72,108,79,101)(73,109,80,102)(74,110,81,103)(75,111,82,104)(76,112,83,105)(77,99,84,106)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,61,26,84)(2,62,27,71)(3,63,28,72)(4,64,15,73)(5,65,16,74)(6,66,17,75)(7,67,18,76)(8,68,19,77)(9,69,20,78)(10,70,21,79)(11,57,22,80)(12,58,23,81)(13,59,24,82)(14,60,25,83)(29,95,55,102)(30,96,56,103)(31,97,43,104)(32,98,44,105)(33,85,45,106)(34,86,46,107)(35,87,47,108)(36,88,48,109)(37,89,49,110)(38,90,50,111)(39,91,51,112)(40,92,52,99)(41,93,53,100)(42,94,54,101), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,45,26,33)(2,44,27,32)(3,43,28,31)(4,56,15,30)(5,55,16,29)(6,54,17,42)(7,53,18,41)(8,52,19,40)(9,51,20,39)(10,50,21,38)(11,49,22,37)(12,48,23,36)(13,47,24,35)(14,46,25,34)(57,110,80,89)(58,109,81,88)(59,108,82,87)(60,107,83,86)(61,106,84,85)(62,105,71,98)(63,104,72,97)(64,103,73,96)(65,102,74,95)(66,101,75,94)(67,100,76,93)(68,99,77,92)(69,112,78,91)(70,111,79,90), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,107,78,100)(72,108,79,101)(73,109,80,102)(74,110,81,103)(75,111,82,104)(76,112,83,105)(77,99,84,106) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,61,26,84),(2,62,27,71),(3,63,28,72),(4,64,15,73),(5,65,16,74),(6,66,17,75),(7,67,18,76),(8,68,19,77),(9,69,20,78),(10,70,21,79),(11,57,22,80),(12,58,23,81),(13,59,24,82),(14,60,25,83),(29,95,55,102),(30,96,56,103),(31,97,43,104),(32,98,44,105),(33,85,45,106),(34,86,46,107),(35,87,47,108),(36,88,48,109),(37,89,49,110),(38,90,50,111),(39,91,51,112),(40,92,52,99),(41,93,53,100),(42,94,54,101)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,80),(58,81),(59,82),(60,83),(61,84),(62,71),(63,72),(64,73),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,45,26,33),(2,44,27,32),(3,43,28,31),(4,56,15,30),(5,55,16,29),(6,54,17,42),(7,53,18,41),(8,52,19,40),(9,51,20,39),(10,50,21,38),(11,49,22,37),(12,48,23,36),(13,47,24,35),(14,46,25,34),(57,110,80,89),(58,109,81,88),(59,108,82,87),(60,107,83,86),(61,106,84,85),(62,105,71,98),(63,104,72,97),(64,103,73,96),(65,102,74,95),(66,101,75,94),(67,100,76,93),(68,99,77,92),(69,112,78,91),(70,111,79,90)], [(1,40,8,33),(2,41,9,34),(3,42,10,35),(4,29,11,36),(5,30,12,37),(6,31,13,38),(7,32,14,39),(15,55,22,48),(16,56,23,49),(17,43,24,50),(18,44,25,51),(19,45,26,52),(20,46,27,53),(21,47,28,54),(57,95,64,88),(58,96,65,89),(59,97,66,90),(60,98,67,91),(61,85,68,92),(62,86,69,93),(63,87,70,94),(71,107,78,100),(72,108,79,101),(73,109,80,102),(74,110,81,103),(75,111,82,104),(76,112,83,105),(77,99,84,106)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | Q8⋊2D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.562+ 1+4 | Dic7⋊4D4 | C22⋊D28 | D14.5D4 | C4⋊D28 | C4⋊C4⋊D7 | C2×D14⋊C4 | C28⋊7D4 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
Matrix representation of C14.562+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 26 | 3 |
| 28 | 2 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 3 | 12 |
| 0 | 0 | 0 | 28 | 5 | 12 |
| 0 | 0 | 1 | 28 | 1 | 0 |
| 0 | 0 | 2 | 22 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 1 | 28 | 0 |
| 0 | 0 | 27 | 7 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 19 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 3 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 14 |
| 0 | 0 | 0 | 0 | 19 | 5 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,0,26,0,0,0,0,10,3],[28,28,0,0,0,0,2,1,0,0,0,0,0,0,28,0,1,2,0,0,0,28,28,22,0,0,3,5,1,0,0,0,12,12,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,28,27,0,0,0,1,1,7,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,9,19,0,0,0,0,14,20,0,0,0,0,0,0,1,3,0,0,0,0,9,28],[12,12,0,0,0,0,0,17,0,0,0,0,0,0,9,15,0,0,0,0,14,20,0,0,0,0,0,0,24,19,0,0,0,0,14,5] >;
C14.562+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{56}2_+^{1+4} % in TeX
G:=Group("C14.56ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1097);
// by ID
G=gap.SmallGroup(448,1097);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations